Here's a partially-filled Hessian matrix. $\begin{bmatrix} -\sin(x) - 2yz & 1 - 2xz & \cos(x) - 2xy \\ \\ 1 - 2xz & 0 & -x^2 \\ \\ ??? & -x^2 & 0 \end{bmatrix}$ What is the missing entry? Choose 1 answer: Choose 1 answer: (Choice A) A $\cos(x) - 2xy$ (Choice B) B $-x^2$ (Choice C) C $0$ (Choice D) D There's not enough information.
Answer: The Hessian of a scalar field $f$ is the matrix that contains all its second-order partial derivative information. $\bold{H}(f) = \begin{bmatrix} f_{xx} & f_{xy} & f_{xz}\\ \\ f_{yx} & f_{yy} & f_{yz} \\ \\ f_{zx} & f_{zy} & f_{zz} \end{bmatrix}$ Because the order of mixed partial derivatives often doesn't matter, the Hessian matrix is usually symmetric. We can use this fact to find $f_{zx}$, which is equal to $f_{xz}$. Matching to the top right corner of the matrix, the missing entry is therefore $\cos(x)-2xy$.